Selected Sampling

Some stakeholders have expressed concerns that the promotional sample contains disproportionate numbers of male members, potentially creating disparate impact issues under anti-discrimination regulations. This analysis applies probability formulas to examine gender representation and explain the discrepancy.

Marginal Probability: Single Event Analysis

Question 1: If a member is randomly selected from the 100,000 pilot participants, what is the probability the member is a woman?

P(Woman from sample) = 43,000 ÷ 100,000 = 0.43 or 43%

Question 2: If a potential customer is randomly selected from UnitedHealth's broader commercially insured membership, what is the probability that person is a woman?

P(Woman from broader membership) = 0.51 or 51% (Kaiser Family Foundation, 2023)

Gap Analysis:

• Absolute gap: 0.51 - 0.43 = 8 percentage points

• Relative underrepresentation: 15.7%

Women are underrepresented by 8 percentage points, representing 15.7% relative underrepresentation compared to the broader population.

Union Probability (Either/Or):

What is the probability a member is either a woman OR enrolled in Medicare Advantage?

P(Woman OR Medicare Adv) = 0.43 + 0.17 - 0.07 = 0.53 or 53%

We subtract the overlap (7,000 members who are both female and in Medicare Advantage) to avoid double-counting.

Joint Probability (And):

What is the probability a member is both a woman AND enrolled in Medicare Advantage?

P(Woman AND Medicare Adv) = 7,000 ÷ 100,000 = 0.07 or 7%

This low joint probability highlights female Medicare Advantage members as a specific demographic requiring targeted outreach.

Conditional Probability (Given That):

Given that a selected member is in Medicare Advantage, what is the probability the member is female?

P(Woman | Medicare Adv) = 0.07 ÷ 0.17 = 0.412 or 41.2%

Among Medicare Advantage members, only 41.2% are women, significantly below the expected 55-57% in the broader Medicare population (Centers for Medicare & Medicaid Services, 2023). This conditional probability reveals Medicare Advantage as a key contributor to overall female underrepresentation.

Statistical Significance

A one-proportion z-test confirms the difference is statistically significant:

z = (0.43 - 0.51) ÷ √[0.51(0.49)/100,000] = -50.6

The z-score of -50.6 far exceeds the critical value of ±1.96 required for 95% confidence (Bluman, 2018). This proves the underrepresentation is not due to random chance and requires corrective intervention.

Gender Distribution Comparison: Pilot Sample vs. Broader Membership

Figure 1: Gender Distribution Comparison: Pilot Sample vs. Broader Membership

The figure above displays the 8 percentage point gap between pilot sample (43% female, 57% male) and broader membership (51% female, 49% male). The visual clearly demonstrates that male representation is elevated in the pilot sample while female representation is suppressed compared to the broader commercially insured population.

Factors Explaining the Discrepancy

Industry Concentration: The sample emphasizes employer plans concentrated in technology, manufacturing, and financial services—industries with higher male workforce participation (U.S. Bureau of Labor Statistics, 2024).

Medicare Advantage Demographics: The pilot's Medicare Advantage segment shows only 41% female representation compared to expected 55-57% in broader Medicare populations (Centers for Medicare & Medicaid Services, 2023).

Primary Policyholder Status: The categorization reflects primary subscribers rather than total covered lives. Historically, primary subscribers were predominantly male. Including dependents would increase female representation by 5-7 percentage points (Claxton et al., 2021).

Plan Type Selection: Women demonstrate higher enrollment in PPO plans, while men show higher representation in high-deductible health plans. The sample may have drawn disproportionately from HDHP populations (Barnett & Berchick, 2017).

Participation Forecasting: Women demonstrate 15-25% higher voluntary

participation rates in wellness programs (Mercer, 2022). The sample may reflect anticipated engagement patterns rather than full eligible population demographics.